Expansion (geometry)

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An example of expanding pentagon into a decagon by moving edges away from the center and inserting new edges in the gaps. The expansion is uniform if all the edges are the same length.
Animation showing an expanded cube (and octahedron)

In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements (vertices, edges, etc.). Equivalently this operation can be imagined by keeping facets in the same position but reducing their size.

The expansion of a regular polytope creates a uniform polytope, but the operation can be applied to any convex polytope, as demonstrated for polyhedra in Conway polyhedron notation. For polyhedra, an expanded polyhedron has all the faces of the original polyhedron, all the faces of the dual polyhedron, and new square faces in place of the original edges.

Expansion of regular polytopes[edit]

According to Coxeter, this multidimensional term was defined by Alicia Boole Stott[1] for creating new polytopes, specifically starting from regular polytopes to construct new uniform polytopes.

The expansion operation is symmetric with respect to a regular polytope and its dual. The resulting figure contains the facets of both the regular and its dual, along with various prismatic facets filling the gaps created between intermediate dimensional elements.

It has somewhat different meanings by dimension. In a Wythoff construction, an expansion is generated by reflections from the first and last mirrors. In higher dimensions, lower dimensional expansions can be written with a subscript, so e2 is the same as t0,2 in any dimension.

By dimension:

  • A regular {p} polygon expands into a regular 2n-gon.
  • A regular {p,q} polyhedron (3-polytope) expands into a polyhedron with vertex figure p.4.q.4.
    • This operation for polyhedra is also called cantellation, e{p,q} = e2{p,q} = t0,2{p,q} = rr{p,q}, and has Coxeter diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png.
      Cube cantellation sequence.svg
      For example, a rhombicuboctahedron can be called an expanded cube, expanded octahedron, as well as a cantellated cube or cantellated octahedron.
  • A regular {p,q,r} polychoron (4-polytope) expands into a new polychoron with the original {p,q} cells, new cells {r,q} in place of the old vertices, p-gonal prisms in place of the old faces, and r-gonal prisms in place of the old edges.
    • This operation for polychora is also called runcination, e{p,q,r} = e3{p,q,r} = t0,3{p,q,r}, and has Coxeter diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png.
  • Similarly a regular {p,q,r,s} polyteron (5-polytope) expands into a new polyteron with facets {p,q,r}, {s,r,q}, {p,q}×{ } prisms, {s,r}×{ } prisms, and {p}×{s} duoprisms.
    • This operation is called sterication, e{p,q,r,s} = e4{p,q,r,s} = t0,4{p,q,r,s} = 2r2r{p,q,r,s} and has Coxeter diagram CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.png.

The general operator for expansion of a regular n-polytope is t0,n-1{p,q,r,...}. New regular facets are added at each vertex, and new prismatic polytopes are added at each divided edge, face, ... ridge, etc.

See also[edit]

Notes[edit]

  1. ^ Coxeter, Regular Polytopes (1973), p. 123. p.210

References[edit]

Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
CDel node 1.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Uniform polyhedron-43-t0.png Uniform polyhedron-43-t01.png Uniform polyhedron-43-t1.png Uniform polyhedron-43-t12.png Uniform polyhedron-43-t2.png Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-33-t0.png Uniform polyhedron-43-h01.png Uniform polyhedron-43-s012.png
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}